MATH 5a
Extra Final Review Problems for Math 5a
1. Simplify completely. In (b), your answer should be in radical notation.
√
(x5 z −2 )2
12 √
(a)
(b)
+ 27
−3 3 −2
(x z )
3
√
√
2. Rewrite the expression 3 −8x5 y · x2 y using exponential notation, then simplify as
much as possible. Leave your answer in exponential notation.
3. Solve the following for x. In (c), write your answer in interval notation.
x−5
(a) x2 + 7x + 1 = 0
(b) x4 = 25x2
(c) 2
≥0
x −4
1
−a
a
4. Simplify completely:
1
+1
a
√
5. Find the domain of f (x) = x2 ex − 2ex .
6. Let f (x) =
1
f (x + h) − f (x)
. Find
and simplify completely.
4x
h
7. The base and top of a cylindrical container are made of high-quality plastic that costs
$7 per square inch. The rest of the container (that is, the lateral surface) is made of
cheaper plastic that costs $3 per square inch. Suppose that the container holds 40
cubic inches. Express the cost of the container as a function of the radius of the base.
8. At 1:00 pm a jeep, travelling at 75 mph, passes an intersection and continues driving
on towards the town of Woodstock, which is 200 miles due west of the interection. At
3:00 pm, an SUV passes through the same intersection heading due south at 60 mph.
(a) Express the distance between the jeep and Woodstock as a function of the time
that has passed since 1:00 pm.
(b) Express the distance between the jeep and the SUV as a function of the time that
has passed since 3:00 pm.
9. A large koi pond is filled from a garden hose at the rate of 10 gal/min. Initially, before
the hose is turned on, the pond contains 300 gallons of water.
(a) Express the volume of water in the pond as a linear function of the time that’s
passed since the hose was turned on.
(b) If the pond has a capacity of 1300 gal, how long does it take to completely fill the
pond with water?
10. Consider the following four functions:
f (x) = cos(πx),
g(x) = ln x,
h(x) = ex ,
w(x) =
1
.
ex
(a) Find the domain of each of these functions.
(b) Find the following and simplify:
(i) (f ◦ g)(e−2 )
(ii) (h + w)(0)
(iii) (f ◦ w)(ln 2)




−
1
if x ≤ −1
x
11. Make a careful and accurate sketch of the graph of f (x) =
.


−x3 + 1, if −1 < x ≤ 1

 √
2 x − 1 if x > 1
12. Let h(x) = 3tan(2x) .
(a) Find two functions f (x) and g(x) such that (f ◦ g)(x) = h(x). Note: You may
not use the functions f (x) = x or g(x) = x.
(b) For what value(s) of x, if any, is h(x) = 0?
−x
e
− 1, if x ≤ 0
. Label
2 ln x,
if x > 0
at least three points on your graph with their exact coordinates.
13. Make an accurate sketch the graph of the function f (x) =
14. Solve the following equations and inequalities for x. In (e) and (f), write your answers
in interval notation.
(c) 3e2x + 17ex = 6
(b) ln x + ln(x − 1) − ln 12 = 0
(a) 3 ln x = 0
4ex
=0
(d) 2
x − 5x + 6
y = ( 2 if ! 0.1
x x<+ 1)0.1
(e) (ln x <
− 4)(ln
≤0 )
(f) x2 ln x + 3x2 < 0
y = ( ! 1 if ! 0.1 < x < 0.1 )
15. Suppose tan θ = − 31 and that csc θ > 0. Find sin θ and cos θ.
16. Find the following. If an expression doesn’t exist, say so. All angle measures are in
radians unless otherwise indicated.
y(a) =cos ( ! 2 if(b)!tan(−
0.1) < x <(c) csc0.1
315 )
◦
5π
2
4π
3
(d) sec π6
17. In each of the following, find the values of θ (if any) in [0, 2π] which make the equation
true. Note: Your answers must be in radians.
√
(a) 2 cos θ = −1
(b) sin2 θ = 34
y = ( 3 if ! 0.1 < x < 0.1 )
y = ( ! 3 if ! 0.1 < x < 0.1 )
18. Make an accurate sketch of the function f (x) = 2 sin(x − π2 ) + 1 on the axes below.
Note: Your graph should extend from x = −2π to x = 2π.
3
2
1
− 5π
2
−2π
− 3π
2
−π
− π2
π
2
−1
−2
−3
π
3π
2
2π
5π
2